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Math 586 Algebraic Combinatorics Presentation Topics and Problems Instructor: Alexander Yong This Document Will Be Updated Throughout the Semester

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Math 586 Algebraic Combinatorics Presentation Topics and Problems Instructor: Alexander Yong This Document Will Be Updated Throughout the Semester Math 586 Algebraic combinatorics Presentation topics and problems Instructor: Alexander Yong This document will be updated throughout the semester. 1. Presentation topics Here are some possible topics. I am happy to make some suggestions about papers to look at to get you started. A presentation will be an explanation of the problem and main re- sults, as well as some computation of data. 1. Mobius inversion and the incidence algebra. 2. The combinatorial nullstellensatz (selected by Emily Heath). 3. The Jacobi-triple product identity 4. Geometric complexity theory. (Selected by William Linz) 5. Brill-Noether theory and set-valued tableaux. 6. Tropical Schur polynomials 7. K-orbit polynomials (e.g., involution Schubert polynomials) 8. Hessenberg varieties and Stanley's chromatic symmetric polynomial (Selected by Harshit Yadav) 9. CSM classes and the Aluffi-Mihalcea conjecture. 10. Monte Carlo methods and algebraic combinatorics. 11. Symmetric group characters as symmetric functions (Orellana-Zabrocki). Note, this topic is not asking about the textbook representation theory statement, but rather something novel due to the two listed mathematicians. (Selected by Simon Lin.) 12. Generalized permutahedra. (Selected by Nicole Yamzon.) 13. Matroid invariants. (Selected by Ramon Garcia Alvarez) 14. The amplituhedron (note the colloquium talk by Hugh Thomas) 15. Total positivity and the Grassmannian. (Selected by Gidon Orelowitz.) 16. \The Complexity of Generating Functions for Integer Points in Polyhedra and Beyond" by Alexander Barvinok (selected by Colleen Robichaux). 17. Applications of Lagrange inversion. 18. Asympototics of generating series. 19. Theory of species (generating series) 20. Tableau Kombinatorics (K-theory); we will partly cover this in class, but there's much more to talk about. This is NOT the same \K" as in \K-orbit". (Selected by Dania Morales). 21. \Subtraction-free complexity, cluster transformations, and spanning trees". (Selected by Thien Le). 2 22. Total positivity: tests and parameterizations. (Selected by Steven Zhang). 23. Noncommutative Schur functions (see papers of Fomin-Greene and Fomin-Blasiak). 24. Puzzle combinatorics. (Selected by Hsin-Po Wang.) 2. Problems for homework Turn in your favorites (at most 10) by the end of the semester. 1. Give a certificate for nonvanishing of the Kostka coefficient Kλ,µ that is polynomial time in the input size of λ, µ (as expressed as bits). This proves nonvanishing of Schurs is in NP. 2. Let Y aδn := xi − xj: 1≤i<j≤n Prove that if ak is not SNP then ak is not SNP. δn δn+1 k 3. Prove that if k is odd then there exists N0 2 N such that aδn for all n ≥ N0. (This exercise depends on Q2.) 4. (Open) Do Q3, except for k even. 5. Prove or disprove: If f; g 2 C[x1; : : : ; xn] are SNP then f · g is SNP. 6. (Open) Prove or disprove Monical's conjecture: if χG is Schur posi- tive then χG is SNP. 7. (Easy) Fix a positive integer n. Lehmer's theorem states that if there is an integer a 2 Z such that an−1 ≡ 1(mod n) and for every prime factor p of n − 1 we have n−1 a p 6≡ 1(mod n) then n is prime. Use this to obtain Pratt's certificate that shows primality is in NP. 8. (Easy) The complement P of a decision problem P is the problem where yes and no are reversed. By definition, P 2 coNP if P 2 NP. Prove that if coNP contains an NP-complete problem, then NP = coNP. 9. Prove that 2 · 5n + (3 + 4i)n + (3 − 4i)n is a positive integer. 10. (I believe this is open): Does there exist a combinatorial rule for the sequence from question 9? 3 11. Prove the following generating series identity 1 1 j Y k X j j Y 1 1 + zx = z x(2) · : 1 − xt k=0 j=0 t=1 Hint: interpret both sides as counting integer partitions of a certain kind, and look for a combinatorial decomposition using a \maximal triangle". 12. (Easy) Prove that f : A ! B and g : B ! A are such that g◦f = idA and f ◦g = idB then f and g are mutually inverse functions. 13. (Straightforward) Complete the bijective proof started in class that the number of labelled trees on n vertices is nn−2. 14. Let A be the set of permutations π 2 Sn such that π(i) 6= i + 1 for 1 ≤ i ≤ n − 1. Let B be the set of permutations that avoid i i + 1 appearing in the one line notations. Give a bijective proof that A ∼= B. 15. (Straightforward) Complete the bijective proof of the Schensted correspondence discussed in class. 16. Let Sch be the Schensted map, whose input is a permutation σ, and output is a pair of tableaux (T;U) of common shape λ. Prove that λ1 = LIS(σ) (longest increasing subsequence length) and `(λ) = LDS(σ) (longest decreasing subsequence length). 17. Prove that if Sch(σ) = (T;U) then Sch(σ−1) = (U; T ). 18. (Straightforward) Complete D´enesbijective proof that ∼ Sn−1 × Tn = Sn−1 × Fn where Tn is the number of labelled trees on n vertices and Fn is the number of factorizations of the cycle (1; 2; 3; : : : ; n) into n − 1 transpo- sitions. 19. Let Fn be as in Q16. Describe a connected graph structure on Fn by giving a \move" between some factorizations. ∼ 20. Give a bijective proof that Tn = Fn that in particular explains the numerical relation between the number of leaves in a tree and the number of factors of the form (i i + 1) (here we interpret n + 1 to mean 1). 21. Give a bijective proof of the hook-length formula λ Y f = n!= hx x where hx is the hook-length of a cell x 2 λ. 4 22. Give a bijective proof that for any fixed λ, that SYT(λ) ∼= BT(λ) where SYT(λ) is the set of Standard Young Tableau of shape λ and BT(λ) is the set of Balanced Tableau of shape λ. 23. Complete the bijective proof that the number of reduced words of w0 (the longest length permutation of Sn) equals SYT(λ0) where λ0 = (n − 1; n − 2;:::; 3; 2; 1). 24. Give a bijection between commutation classes of reduced words of w0 2 Sn and rhombic tilings of a regular 2n-gon. 25. Let Λ = f(i; j) : 1 ≤ i ≤ k; 1 ≤ j ≤ ng: Fix a Young diagram λ ⊆ Λ. Then let A = f(i; j) : 1 ≤ i ≤ k; λ1 − λi + 1 ≤ j ≤ n + λ1 − λig and B = f(i; j): k + 1 ≤ i ≤ 2k; n + λ1 − λi−k + 1 ≤ j ≤ n + λ1g: Set θ = A [ B. Give a bijective proof that Hooklengths(λ) [ Hooklengths(Λ) = Hooklengths(θ); where we mean multiset union and equality. For example, suppose λ = ; Λ = then θ = : One can check that Hooklengths(λ) = f12; 3g; Hooklengths(Λ) = f1; 22; 33; 43; 52; 6g and Hooklengths(θ) = f13; 22; 34; 43; 52; 6g: 26. Prove that the Symmetric group Sn is isomorphic to the free 2 group generated by s1; s2; : : : ; sn−1 modulo the relations si = ;, the commutation relation sisj = sjsi (for ji−jj ≥ 2), and the braid relation sisi+1si = si+1sisi+1. 5 27. Prove J. Tits' lemma: any two reduced words for w 2 Sn can be obtained from one another by a sequence of commutation and braid relations (see Q26). Prove that any two reduced words have the same length `(w), where `(w) is the number of inversions of w. 28. (Open) Give a \nicer" and direct proof of the hook-length formula, as a count of Balanced tableau of a shape λ. 29. Let Hooklengths(λ) be the multiset of hook-lengths of λ. No- tice that if λ0 is the conjugate shape of λ then Hooklengths(λ) = Hooklengths(λ0). Prove or disprove: if λ, µ are Young diagrams such that Hooklengths(λ) = Hooklengths(µ) (multiset equality), then µ = λ or µ = λ0. 30. (a) Prove that dominance order (Parn; λD) is self-dual under con- jugation. (b) Prove that dominance order is a lattice. 31. Recall that if λ = (λ1 ≥ · · · ≥ λn) is a partition, then the λ- permutahedron is the convex hull of the Sn orbit of λ. Prove that µ ≤D λ if and only if Pµ ⊆ Pλ. 32. Let Nλ be the set of Nilpotent matrices of Jordan type λ. Prove that µ ≤D λ if and only if Nµ is contained in the closure of Nλ ⊂ Matn×n, with respect to the standard topology. 33. Recall the AGM inequality: for nonnegative reals x1; : : : ; xn, x + ··· x 1 n ≥ (x x ··· x )1=n: n 1 2 n This exercise concerns a generalization. For λ 2 Par(n) define the λ- mean of x ; : : : ; x 2 to be 1 P w(xλ). Prove that λ-mean≥ 1 n R≥0 n! w2Sn µ-mean if and only if λ ≥ µ. 34. Prove that n n−1+r X xk hr(x1; : : : ; xn) = Q : (xk − xi) k=1 i6=k 35. Let Fλ = !(mλ) be the forgotten symmetric function. Determine the expansion of Fλ in terms of the fmµg-basis. 36. Prove that X 1 X 1 1 3 2n − 1 = = · ··· : zλ zλ 2 4 2n λ`2n; all parts even λ`2n; all parts odd 6 37. Give a direct combinatorial proof that zλ is the number of elements that commpute with wλ, a permutation of cycle type λ. 38. Let Πn ⊆ Λn consist of symmetric functions f of degree n satisfying the cancellation law f(x1; −x1; x3; x4;:::) = f(x3; x4;:::): n n Prove that Π = (Q[p1; p3; p5;:::]) .
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